X.O.
Ok, the easy issue first - YUCK, I would NEVER in a billion years want the engagement ring of a woman my fiance was married to, regardless of the reason they are no longer married! Not romantic. A bit morbid. Yes, the rings of his deceased wife should go to her children or grandchildren, not to his new wife. The engagement ring doesn't have to be flashy, but he really should get a new and special ring for his fiance.
My aunt's husband passed away VERY unexpectedly about 10 years ago (as in, they were talking, and suddenly he just died). Within 2 years she was married to a long time friend whose spouse had also passed away. It was extremely difficult for my cousin to accept the fact that her mother was married to a man, other than her father. She felt it was disrespectful to her dad's memory. She felt that her mother jumped the gun. She felt that she was being cast aside. 10 years later, she has finally accepted the marriage. I barely know my aunt's new husband, as I've been living away from my hometown this whole time. Still, when I do see him I treat him just as I used to treat my uncle--I make small talk, and try to get to know him a bit, and be as polite as I can. I don't have an emotional connection to him, but you better believe I am pleased as pie that my aunt found someone to help heal her broken heart. Her daughter is FINALLY happy for her mom.
My grandmother remarried 7 years after my grandfather passed away. Her new husband only lived about 6 months after they married (in their early 70s), but they didn't want to waste any time. So glad that they didn't wait. He made my grandma so happy, and she was his dream woman.
I say, butt out about the relationship, but gently suggest that she deserves her own unique ring.